1. Heat convection problem : In order to investigate the global structure of the solution space of the nonlinear PDE's and to treat the global bifurcation curves in it, we worked on the analytical method combined with the computational analysis and computer assisted proof. We proposed criterions to prove the existence of solutions which correspond to parameter values as computer assited proof. Using the method we showed the existence of global bifurcation curves on which the roll-type solutions exist that correspond to large Rayleigh numbers. In the case of 3-dimension we investigated numerically the pattern formation of roll-type, rectangle-typpe and hexagonaltype solutions and their stability, and we clarified the global bifurcation diagram which is not seen from the local bifurcation theory.2. Taylor problem : We considered the stability of Couette flow when the two cylinder rotate in the opposite directions. It is reduced to the eigenvalue problem for the system of ordinary differe
… Morential equations and it can be treated by our computer assisted proof to see the exact critical Taylor number, at which the stationary or Hopf bifurcation occurs. The bifurcation point with multiplicity is one of our future subject.3. The existence theorem for stationary solution of Navier-Stokes equation is proved by our numerical verification method at least for small Reynols number.4. Dynamical systems : We know that when the degeneracy of singular points of vector field increases, the behavior of dynamics becomes more complex and the global phenomena become more included. We investigated the singular point with codimension 3 and proved analytically that the hetero-clinic cycle bifurcates and also chaotic attractor does.5. For the 3-dimensional exterior problem of stationary Navier-Stokes equation, we introduced a real interpolation of Morrey spaces to solve N-S equation and succeeded to construct the exterior stationary solution and to prove its stability without the unnatural zero net force conditions. Less